In Fig. REF  (Top), we first fix the number of machines \(M = 10\) , and the number of total synchronization steps to be 100, and vary the number of local iterations between two synchronization steps, i.e., \(h \in \lbrace 1, 10, 25, 50, 100, 200\rbrace .\) 
Increasing \(h\) , i.e., each distributed machine performing more local iterations, leads to faster convergence in terms of the synchronization steps. Notably, the speed up gets marginal: e.g., there is not much difference between \(h=100\)  and \(h=200\) , indicating there is an “optimal” \(h\)  that leads to the biggest reduction in the number of synchronization steps.
Note that \(\varepsilon \)  does not decrease below certain level due to the inherent finite sampling error of quantum measurements [1]}.
